The axiom of extensionality


Axiom of extensionality

If two sets contain the same elements, they are equal.

\(\forall x \forall y[\forall z(z\in x \leftrightarrow z\in y)\rightarrow x=y]\)

This is an axiom, not a definition, because equality was defined as part of first-order logic.

Note that this is not bidirectional. \(x=y\) does not imply that \(x\) and \(y\) contain the same elements. This is appropriate as \(\dfrac{1}{2}= \dfrac{2}{4}\) for example, even though they are written differently as sets.

Reflexivity of equality

The reflexive property is:

\(\forall x(x=x)\)

We can replace the instance of \(y\) with \(x\):

\(\forall x [\forall z(z\in x \leftrightarrow z\in x)\rightarrow x=x]\)

We can show that the following is true:

\(\forall z(z\in x \leftrightarrow z\in x)\)


\(\forall x [T \rightarrow x=x]\)


Symmetry of equality

The symmetry property is:

\(\forall x \forall y[(x=y)\leftrightarrow (y=x)]\)

We know that the following are true:

\(\forall x \forall y[\forall z(z\in x \leftrightarrow z\in y)\rightarrow x=y]\)

\(\forall x \forall y[\forall z(z\in x \leftrightarrow z\in y)\rightarrow y=x]\)


\(\forall x \forall y[\forall z(z\in x \leftrightarrow z\in y)\rightarrow (x=y\land y=x)]\)

Transitivity of equality

The transitive property is:

\(\forall x \forall y \forall z[(x=y \land y=z) \rightarrow x=z]\)

Substitution for functions

The substitutive property for functions is:

\(\forall x \forall y[(x=y)\rightarrow (f(x)=f(y))]\)

Substitution for formulae

The substitutive property for formulae is:

\(\forall x \forall y[((x=y)\land P(x))\rightarrow P(y)]\)

Doesn’t this require iterating over predicates? Is this possible in first order logic??

Result 1: The empty set is unique

We can now show the empty set is unique.

Result 2: Every element of a set exists

If an element did not exist, the set containing it would be equal to a set which did not contain that element.

Result 3: Sets are unique

Equivalence classes

We have already ready defined the relationship equality, between terms.


Sometimes we may wish to talk about a collection of terms which are all equal to each other. This is an equivalence class.

Though we have not yet defined them, integers are example of this. For example \(-1\) can be written as \(0-1\), \(1-2\) and so on.

\(\forall y \forall x x=y\rightarrow x\in z\)

For all sets, we can call the class of all sets equal to the set an equivalence class.

This does not necessarily exist.